The local spectrum of the Dirac operator for the universal cover of $S{L}_2(\mathbb{R})$

Abstract

Using representation theory, we compute the spectrum of the Dirac operator on the universal covering group of $S{L}_2(\mathbb{R})$, exhibiting it as the generator of $K{K}^1(\mathbb{C},\mathfrak{A})$, where $\mathfrak{A}$ is the reduced $C^{\ast }$-algebra of the group. This yields a new and direct computation of the $K$-theory of $\mathfrak{A}$. A fundamental role is played by the limit-of-discrete-series representation, which is the frontier between the discrete and the principal series of the group. We provide a detailed analysis of the localised spectra of the Dirac operator and compute the Dirac cohomology.

17 pages, 6 figures